stable lévy motion with values in the skorohod space€¦ · multivariate case regular variation...

Stable Levy motion with values in theSkorohod space

Raluca BalanUniversity of Ottawa

Joint work with Becem Saidani

Stable Levymotion with values

in the Skorohodspace

Stable FCLT

Multivariate Case

Regular variationin D

D-valued α-stableLevy motion

Functional limittheorem in D

Table of Contents

1 Stable FCLT

2 Multivariate Case

3 Regular variation in D

4 D-valued α-stable Levy motion

5 Functional limit theorem in D



Stable FCLT

Multivariate Case




1. Stable Functional CLT

Theorem 1 (Skorohod, 1957)

(Xi )i≥1 i.i.d. with regularly varying tails of index α ∈ (0, 2):

P(|X | > x) = x−αL(x), limx→∞

P(X > x)

P(|X | > x)= p ∈ [0, 1],

where L is slowly varying at ∞. If Sn(t) =∑[nt]

i=1 Xi , then1

an(Sn(t)− bn(t))

t≥0

d→ Z (t)t≥0 in D([0,∞)),

where Z (t)t≥0 is an α-stable Levy motion, aαn ∼ nL(an),bn(t) = 0 if α < 1 and bn(t) = E [Sn(t)] if α > 1.



Stable FCLT

Multivariate Case




Properties of α-stable Levy motion:

(i) Z (t2)− Z (t1), . . . ,Z (tn)− Z (tn−1) are independent, forany t1 < t2 < . . . < tn

(ii) Z (t2)− Z (t1)d= Z (t2 − t1) for any t1 < t2

(iii) Z (t) ∼ Sα(t1/ασ, β, 0) with β = p − q, q = 1− p and

σα = Γ(2−α)1−α cos

(πα2

)=: C−1

α . We have:

E (e iuZ(t)) = exp

t

∫R

(e iuy − 1)να,p(dy)

, if α < 1

E (e iuZ(t)) = exp

t

∫R

(e iuy − 1− iuy)να,p(dy)

, if α > 1

να,p(y) =(pαy−α−11y∈(0,∞)+qα(−y)−α−11y∈(−∞,0)

)dy



Stable FCLT

Multivariate Case




Idea of proof α < 1 (Resnick, 1986)

1) Point process convergence:

Nn =n∑

i=1

δXi/and→ N =

∑i≥1

δJi = PRM(να,p)

follows from nP( Xan∈ ·) v→ να,p in R0 = [−∞,∞]− 0.

2) Continuity of truncated summation: P(N ∈ Λ) = 1∑i≥1

δxi 7→∑i≥1

xi1|xi |>ε continuous on Λ

3) S(ε)n = a−1

n

∑ni=1 Xi1|Xi |>εan

d→ Z (ε) =∑

i≥1 Ji1|Ji |>ε4) Let ε→ 0.



Stable FCLT

Multivariate Case




Idea of proof (functional case)1) Point process convergence:

Nn =∑i≥1

δ( in,Xian

)

d→ N =∑i≥1

δ(Ti ,Ji ) = PRM(Leb × να,p)

2) Continuity of truncated summation: the map

Qε :∑i≥1

δ(ti ,xi ) 7→

∑ti≤t

xi1|xi |>ε

t∈[0,T ]

∈ D([0,T ])

is continuous on a set Λ and P(N ∈ Λ) = 1.3) Define

S(ε)n = Qε(Nn) =

1

an

[nt]∑i=1

Xi1|Xi |>εan

t∈[0,T ]

Z (ε) = Qε(N) =

∑Ti≤t

Ji1|Ji |>ε

t∈[0,T ]



Stable FCLT

Multivariate Case




By Continuous Mapping Theorem, for any ε > 0,

S(ε)n (·) d→ Z (ε)(·) in D([0,T ])

4) Let ε→ 0:

limε→0

lim supn→∞

P( supt∈[0,T ]

|S (ε)n (t)− Sn(t)| > δ) = 0

supt≤T|Z (ε)(t)− Z (t)| → 0 a.s.

The conclusion follows by Theorem 4.2 of Billingsley (1968).

RemarkA similar argument works when α > 1, using centering.



Stable FCLT

Multivariate Case




2. Multivariate Case

A random vector X in Rd is regularly varying (RV) if

nP

(X

an∈ ·)

v→ ν in Rd0 (1)

where ν is a Radon measure on Rd0 = [−∞,∞]d − 0 with

ν(Rd0 − Rd) = 0. In this case, there exists α > 0 s.t.

ν(cA) = c−αν(A), for any c > 0,A ∈ B(Rd).

Moreover, (1) is equivalent to:

nP

(( |X |an,X

|X |

)∈ ·)

v→ cνα × Γ1 in (0,∞]× Sd ,

for some probability measure Γ1 on Sd = x ∈ Rd ; |x | = 1,c > 0 and να(r ,∞) = r−α for r > 0 and να∞ = 0.



Stable FCLT

Multivariate Case




Multivariate Stable FCLT

Theorem 2 (Resnick, 2007)

(Xi )i≥1 i.i.d. in Rd satisfying (1). If Sn(t) =∑[nt]

i=1 Xi , then

1

an(Sn(t)− bn(t))

t≥0

d→ Z (t)t≥0 in D([0,∞);Rd),

where Z (t)t≥0 is an α-stable Levy motion in Rd :

E [e iuZ(t)] = exp

t

∫Rd

(e iu·x − 1)ν(dx)

, if α < 1;

E [e iuZ(t)] = exp

t

∫Rd

(e iu·x − 1− iu · x)ν(dx)

, if α > 1.



Stable FCLT

Multivariate Case




3. Regular variation in D

D = D([0, 1]) is the set of cadlag functions on [0, 1]Xi = Xi (s)s∈[0,1], i ≥ 1 i.i.d. random elements in D

Examples

1) Xi (s) =number of internet transactions on a website onday i at time s during the day2) Xi (s)=high tide water level on day i at location s on theshore of Netherlands (de Haan and Lin, 2001)

GoalStudy the asymptotic behaviour of the partial sum process:

Sn(t, s) =

[nt]∑i=1

Xi (s), t ≥ 0, s ∈ [0, 1]

Note: Sn(·) ∈ D([0,∞);D) where Sn(t) = Sn(t, s)s∈[0,1]



Stable FCLT

Multivariate Case




Basic properties of D (Billingsley 1968, 1999)

Uniform normD is a Banach space equipped with the uniform norm:

‖x‖ = sups∈[0,1]

|x(s)|

(D, ‖ · ‖) is not separable

Skorohod J1-distance

D is a metric space equipped with the Skorohod J1-distance:

dJ1(x , y) = infλ∈Λ‖λ− e‖ ∨ ‖x − y λ‖

The Borel σ-field on (D, J1) coincides with the σ-fieldgenerated by the projections πs : D→ R, πs(x) = x(s).



Stable FCLT

Multivariate Case




Polar coordinate transformationWe consider the map T : D0 → (0,∞)× SD given by:

T (x) =

(‖x‖, x

‖x‖

)where D0 = D− 0 and SD = x ∈ D; ‖x‖ = 1T is a homeomorphism (since ‖ · ‖ is J1-continuous)

The product space

D0 = (0,∞]× SD is a Polish space with distance:

dD0

((r , z), (r ′, z ′)

)=

∣∣∣∣1r − 1

r ′

∣∣∣∣ ∧ d0J1

(z , z ′)

D0 is not locally compact



Stable FCLT

Multivariate Case




Definition (de Haan and Lin, 2001)

A random element X in D is regularly varying if ∃ an →∞,α > 0, c > 0 and a probability measure Γ1 on SD such that

nP

((‖X‖an

,X

‖X‖

)∈ ·)

w→ cνα × Γ1 in D0. (2)

(µnw→ µ if µn(A)→ µ(A) for any A bounded, µ(∂A) = 0)

Let ν be the measure on D0 given by

ν T−1 = cνα × Γ1 =: ν. (3)

Example

If X = X (s)s∈[0,1] is an α-stable Levy motion with samplepaths in D, then X is regularly varying in D.



Stable FCLT

Multivariate Case




4. D-valued stable Levy motion

DefinitionLet ν be a measure on D s.t. ν0 = 0 and (3) holds.For any t > 0, let Z (t) be a random element in D.Z (t)t≥0 is a D-valued α stable Levy motion if Z (0) = 0,(i) Z (t2)− Z (t1), . . . ,Z (tn)− Z (tn−1) are independent

(ii) Z (t2)− Z (t1)d= Z (t2 − t1) for any t1 < t2

(iii) Z (t, s)s∈[0,1] is a cadlag α-stable process s.t. if α < 1,

E [e i∑m

j=1 ujZ(t,sj )] = exp

t

∫Rm

(e iu·y − 1)µs1,...,sm(dy)

,

and if α > 1,

E [e i∑m

j=1 ujZ(t,sj )] = exp

t

∫Rm

(e iu·y − 1− iu · y)µs1,...,sm(dy)

,

whereµs1,...,sm = ν π−1

s1,...,sm .



Stable FCLT

Multivariate Case




Example

Let L(t, s)t,s∈[0,1] be an α-stable Levy sheet withsample paths in D([0, 1]2). Then L(t)t∈[0,1] is aD-valued α-stable Levy motion.

Proof of (iii): L(t) = L(t, s)s∈[0,1] is a cadlag α-stableLevy motion. Then L(1) is RV in D with limiting measurecνα × Γ1 (Hult and Lindskog, 2007). Let ν be given by (3).(L(1, s1), . . . , L(1, sm)) is RV in Rm with limiting measureν π−1

s1,...,sm (Hult and Lindskog, 2005)But (L(1, s1), . . . , L(1, sm)) is α-stable and (if α < 1)

E [e i∑m

j=1 ujL(1,sj )] = exp

∫Rm

(e iu·y − 1)µs1,...,sm(dy)

.

(L(1, s1), . . . , L(1, sm)) is RV in Rm with limiting measureµs1,...,sm . Hence

µs1,...,sm = ν π−1s1,...,sm



Stable FCLT

Multivariate Case




Construction of D-valued stable Levy motion

Compound Poisson building blocks

Let N be a PRM on [0,∞)× D0 of intensity Leb × ν.Let εj ↓ 0 with ε0 = 1. For any t ≥ 0 and s ∈ [0, 1], define

Zj(t, s) =

∫[0,t]×Ij×SD

rz(s)N(du, dr , dz)

where Ij = (εj , εj−1] for j ≥ 1 and I0 = (1,∞). The processZj(t) = Zj(t, s)s∈[0,1] has all sample paths in D and

E (e iuZj (t,s)) = exp

t

∫Ij×SD

(e iurz(s) − 1)ν(dr , dz)

.



Stable FCLT

Multivariate Case




Let ϕ(s) =∫SD z(s)Γ1(dz) and ψ(s) =

∫SD |z(s)|2Γ1(dz).

E (Zj(t, s)) = tϕ(s)

∫Ij

rνα(dr), j ≥ 0

Var(Zj(t, s)) = tψ(s)

∫Ij

r2να(dr), j ≥ 0

Zj(t, s)j≥0 are independent. By Kolmogorov’s criterion,∑j≥1

(Zj(t, s)− E (Zj(t, s))

)converges a.s.

If α < 1, let Z (εk )(t, s) =∑k

j=0 Zj(t, s) and

Z (t, s) =∑j≥0

Zj(t, s).

If α > 1, let Z(εk )

(t, s) =∑k

j=0

(Zj(t, s)− E (Zj(t, s))

)and

Z (t, s) =∑j≥0

(Zj(t, s)− E (Zj(t, s))

)



Stable FCLT

Multivariate Case




Lemma(Z (t, s1), . . . ,Z (t, sm)) has a stable distribution in Rm.

Proof: Assume α < 1. (The case α > 1 is similar.)

E (e iuZ(t,s)) = exp

t

∫D0

(e iurz(s) − 1)ν(dr , dz)

= exp

t

∫R

(e iuy − 1)µs(dy)

where µs = ν π−1

s = ν π−1s . By the scaling property of ν,

µs(cA) = c−αµs(A) for any c > 0,A ⊂ R. Hence,

µs(dy) =(c+s y−α−11y>0 + c−s (−y)−α−11y<0

)dy

and Z (t, s) ∼ Sα(t1/ασs , βs , 0) for some σs > 0, βs ∈ R.



Stable FCLT

Multivariate Case




Case α < 1

Theorem 1.(a) (B-Saidani, 2019)

For any t > 0, there exists a cadlag modificationZ (t) = Z (t, s)s∈[0,1] of Z (t) = Z (t, s)s∈[0,1] such that

limk→∞

‖Z (εk )(t)− Z (t)‖ → 0 a.s.

Z (t)t≥0 is a D-valued α-stable Levy motion.

Proof: ‖Zj(t)‖ ≤∫

[0,t]×Ij×SD rN(du, dr , dz) and

E∑j≥1

‖Zj(t)‖ ≤ t

∫(0,1]×SD

rν(dr , dz) <∞

It follows that∑

j≥1 ‖Zj(t)‖ <∞ a.s. and Z (εk )(t)k isCauchy in (D, ‖ · ‖) a.s. (with limit Z (t)).



Stable FCLT

Multivariate Case




Theorem 1.(b) (B-Saidani, 2019)

There exists a collection Z (t)t≥0 of random elements

in D such that P(Z (t) = Z (t)) = 1 for any t > 0 and

supt≤T‖Z (εk )(t)− Z (t)‖ → 0 ∀T > 0.

Moreover, the map t 7→ Z (t) is in Du([0,∞);D) a.s.

NotationDu([0,∞);D) is the set of cadlag functions x : [0,∞)→ Dwith respect to the uniform norm ‖ · ‖ on D.

RemarkDu([0, 1];D) is a Banach space with respect to thesuper-uniform norm:

‖x‖D = supt≤1‖x(t)‖.



Stable FCLT

Multivariate Case




Case α > 1

Theorem 2.(a) (B-Saidani, 2019)

For any t > 0, there exists a cadlag modificationZ (t) = Z (t, s)s∈[0,1] of Z (t) = Z (t, s)s∈[0,1] such that

limk→∞

‖Z (εk )(t)− Z (t)‖ → 0 a.s. (4)

Z (t)t≥0 is a D-valued α-stable Levy motion.

Proof: To prove (4), we use a version of Ito-Nisio Theorem(Basse O’Connor and Rosinski, 2013):

• Z (εk )(t) =

∑kj=0

(Zj(t)− E (Zj(t))

)is a sum of zero-mean

independent random elements in D• Z (εk )

(t)k is tight in (D, J1) (Roueff and Soulier, 2015)

• Z (εk )(t)

d→ Z (t) in D and Z (t, s)s∈[0,1] is u.i.



Stable FCLT

Multivariate Case




Theorem 2.(b) (B-Saidani, 2019)

There exists a collection Z (t)t≥0 of random elements

in D such that P(Z (t) = Z (t)) = 1 for any t > 0, themap t 7→ Z (t) is in D([0,∞);D) and

Z(εk )

(·) d→ Z (·) in D([0,∞);D)

as k →∞, k ∈ N ′ for a subsequence N ′. D([0,∞);D) is theset of cadlag functions x : [0,∞)→ D with respect to J1.

J1-distance

Here D([0, 1];D) is equipped with J1-distance: (Whitt, 1980)

dD(x , y) = infλ∈Λ‖λ− e‖ ∨ ρD(x , y λ) ,

where ρD is the uniform distance with respect to d0J1

:

ρD(x , y) = supt≤1

d0J1

(x(t), y(t)) ≤ ‖x − y‖D.



Stable FCLT

Multivariate Case




Proof:

• Similarly to Roueff and Soulier (2015), it can be proved

that (Z(εk )

)k is tight in D([0,∞);D). There exists aprocess Y (t)t≥0 with sample paths in D([0,∞);D)(defined on another space (Ω′,F ′,P ′)) such that:

Z(εk )

(·) d→ Y (·), in D([0,∞);D)

as k →∞, k ∈ N ′.

• (Z (t1), . . . ,Z (tn))d= (Y (t1), . . . ,Y (tn)) for all

t1, . . . , tn > 0

• By Lemma 3.24 (Kallenberg, 2002), it follows thatZ (t)t≥0 has a modification Z (t)t≥0 with samplepaths in D([0,∞);D).



Stable FCLT

Multivariate Case




5. Functional limit theorem (B-Saidani, 2019)

Theorem 3.Let (Xi )i≥1 be i.i.d. RV in D:

nP

((‖X‖an

,X

‖X‖

)∈ ·)

w→ cνα × Γ1 in D0.

Let Sn(t, s) =∑[nt]

i=1 Xi (s). If α > 1, assume an additionaltechnical condition. Then

1

an(Sn(t)− bn(t))

t≥0

d→ Z (t)t≥0 in D([0,∞);D),

where Z (t)t≥0 is the process of Theorems 1-2.



Stable FCLT

Multivariate Case




Proof (case α < 1)1) Point process convergence:

Nn =∑i≥1

δ( in,‖Xi‖an

,Xi

‖Xi‖

) d→ N =∑i≥1

δ(Ti ,Ri ,Wi ) = PRM(Leb×ν)

2) Continuity of truncated summation: the map

Qε :∑i≥1

δ(ti ,ri ,zi ) 7→

∑ti≤t

rizi1ri>ε

t∈[0,T ]

∈ D([0,T ];D)

is continuous on a set Λ and P(N ∈ Λ) = 1.3) Define

S(ε)n = Qε(Nn) =

1

an

[nt]∑i=1

Xi1‖Xi‖>εan

t∈[0,T ]

Z (ε) = Qε(N) =

∑Ti≤t

RiWi1Ri>ε

t∈[0,T ]



Stable FCLT

Multivariate Case




By Continuous Mapping Theorem, for any ε > 0,

S(ε)n (·) d→ Z (ε)(·) in D([0,T ];D)

4) Let ε→ 0:

limε→0

lim supn→∞

P( supt∈[0,T ]

‖S (ε)n (t)− Sn(t)‖ > δ) = 0

supt≤T‖Z (εk )(t)− Z (t)‖ → 0 a.s. (Theorem 1.(b))

The conclusion follows by Theorem 4.2 of Billingsley (1968).

RemarkA similar argument works when α > 1, using centering.



Stable FCLT

Multivariate Case




RemarkTheorem 3 extends a result of Roueff and Soulier (2015) tofunctional convergence. Their result says that if α > 1 and(Xi )i≥1 are i.i.d. regularly varying in D, then

1

an

n∑i=1

(Xi − E (Xi )

) d→ Y in D,

where Y = Y (s)s∈[0,1] is an α-stable process with samplepaths in D. Note that

Y (s)s∈[0,1]d= Z (1, s)s∈[0,1].



Stable FCLT

Multivariate Case




References

• Balan, R.M. and Saidani, B. (2019). Stable Levymotion with values in the Skorohod space: constructionand approximation. To appear in J. Theor. Probab.Preprint arXiv:1809.02103

• Basse-O’Connor, A. and Rosinski, J. (2013). On theuniform convergence of random series in Skorohodspace and representation of cadlag infinitely divisibleprocesses. Ann. Probab. 41, 4317-4341.

• Billingsley, P. (1968, 1999). Convergence ofProbability Measures. John Wiley.



Stable FCLT

Multivariate Case




• de Haan, L. and Lin, T. (2001). On convergencetowards an extreme value distribution in C [0, 1]. Ann.Probab. 29, 467-483.

• Hult, H. and Lindskog, F. (2005). Extremal behaviourof regularly varying stochastic processes. Stoch. Proc.Appl. 115, 249-274

• Hult, H. and Lindskog, F. (2007). Extremal behaviourof stochastic integrals driven by regularly varying Levyprocesses. Ann. Probab. 35, 309-339.

• Kallenberg, O. (2002). Foundations of ModernProbability. Second edition. Springer.

• Resnick, S.I. (2007). Heavy-Tail Phenomena:probabilistic and statistical modelling. Springer.



Stable FCLT

Multivariate Case




• Roueff, F. and Soulier, P. (2015). Convergence tostable laws in the space D. J. Appl. Probab. 52, 1-17.

• Skorohod, A. V. (1957). Limit theorems for stochasticprocesses with independent increments. Th. Probab.Appl. 2, 138-171.

• Whitt, W. (1980). Some useful functions for functionallimit theorems. Math. Oper. Res. 5, 67-85.



Stable FCLT

Multivariate Case



Functional limittheorem in D Thank you!

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